In this paper, we propose to build flows on compact Riemannian manifolds following the celebrated theory of [1]}, that is using exponential map applied to gradients of \(c\) -concave functions. Our main contribution is observing that the rather intricate space of \(c\) -concave functions over arbitrary compact Riemannian manifold can be approximated with discrete \(c\) -concave functions. We provide a theoretical result showing that discrete \(c\) -concave functions are dense in the space of \(c\) -concave functions. We use this theoretical result to prove that maps defined via a discrete \(c\) -concave potentials are universal. Namely, can approximate arbitrary optimal transports between an absolutely continuous source distribution and arbitrary target distribution on the manifold.
